Properties

Label 2-33-33.20-c2-0-1
Degree $2$
Conductor $33$
Sign $0.815 - 0.578i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 + 0.619i)2-s + (−1.63 + 2.51i)3-s + (0.0135 + 0.00987i)4-s + (5.21 − 1.69i)5-s + (−4.67 + 3.78i)6-s + (−4.52 − 3.28i)7-s + (−4.69 − 6.45i)8-s + (−3.66 − 8.22i)9-s + 10.9·10-s + (2.23 + 10.7i)11-s + (−0.0470 + 0.0180i)12-s + (−3.00 + 9.24i)13-s + (−6.58 − 9.06i)14-s + (−4.25 + 15.8i)15-s + (−4.96 − 15.2i)16-s + (16.9 − 5.52i)17-s + ⋯
L(s)  = 1  + (0.953 + 0.309i)2-s + (−0.544 + 0.838i)3-s + (0.00339 + 0.00246i)4-s + (1.04 − 0.338i)5-s + (−0.778 + 0.630i)6-s + (−0.646 − 0.469i)7-s + (−0.586 − 0.807i)8-s + (−0.406 − 0.913i)9-s + 1.09·10-s + (0.203 + 0.979i)11-s + (−0.00392 + 0.00150i)12-s + (−0.230 + 0.710i)13-s + (−0.470 − 0.647i)14-s + (−0.283 + 1.05i)15-s + (−0.310 − 0.955i)16-s + (0.999 − 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.815 - 0.578i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.25316 + 0.399478i\)
\(L(\frac12)\) \(\approx\) \(1.25316 + 0.399478i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 - 2.51i)T \)
11 \( 1 + (-2.23 - 10.7i)T \)
good2 \( 1 + (-1.90 - 0.619i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (-5.21 + 1.69i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (4.52 + 3.28i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (3.00 - 9.24i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-16.9 + 5.52i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (15.0 - 10.9i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 12.3iT - 529T^{2} \)
29 \( 1 + (1.45 - 2.00i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-15.2 + 46.8i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-31.8 - 23.1i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (33.2 + 45.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 43.9T + 1.84e3T^{2} \)
47 \( 1 + (-33.9 - 46.7i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-41.0 - 13.3i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-52.9 + 72.8i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (9.53 + 29.3i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 34.0T + 4.48e3T^{2} \)
71 \( 1 + (35.7 - 11.6i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-9.81 - 7.12i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (19.5 - 60.0i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (9.22 - 2.99i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 34.1iT - 7.92e3T^{2} \)
97 \( 1 + (11.6 - 35.9i)T + (-7.61e3 - 5.53e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.59960070120143384693711713047, −15.27092670358914085928211681759, −14.27412412202267055876902780684, −13.16408443771375591747199046727, −12.02081740028019086296148933310, −9.955695616487394603968570102368, −9.548048878464864925281611963152, −6.56955743871093617745612669867, −5.40887361338099353619873721221, −4.06239140290557108042790310609, 2.80683410649141799733119501045, 5.46674441313917918928535994250, 6.35378772470517259878756688678, 8.531682389078934187185456515212, 10.41419738038276192292633382193, 11.88943698816893532297853039156, 12.90471829486430214856393067692, 13.62179476106867535234581549027, 14.70465867790606122315194338825, 16.60387930948094253381705629391

Graph of the $Z$-function along the critical line